Power Control in an Optical Fiber Network

ABSTRACT

Multiple receivers are comprised in a flexible coherent transceiver of a multi-span optical fiber network. Each of the multiple receivers is operative to handle communications on a respective channel. The multiple receivers measure optical characteristics. For each of the multiple receivers, the optical characteristics include optical nonlinear interactions on the respective channel, the optical nonlinear interactions being at least partially dependent from one span to another span. An optical power of a signal on each of the multiple channels is adjusted as a function of the optical characteristics.

CROSS-REFERENCE

This application is a continuation-in-part of U.S. patent applicationSer. No. 15/603,810 filed May 24, 2017, the contents of which areincorporated herein by reference.

This application is related to U.S. patent application Ser. No.15/648,895 filed Jul. 13, 2017, the contents of which are incorporatedherein by reference. U.S. patent application Ser. No. 15/648,895 is acontinuation of U.S. patent application Ser. No. 15/603,810.

TECHNICAL FIELD

This document relates to the technical field of optical communicationsand specifically to the control of components in an optical fibernetwork.

BACKGROUND

Current best practices for determining optical parameters in an opticalfiber network look at equalizing the ratio of amplified spontaneousemission (ASE) to signal power on channels over an optical section whilerespecting channel power limits to manage the fiber optical nonlineareffects. This equalization addresses the strong power tilt that canaccumulate across spans of optical fiber mainly due to Stimulated RamanScattering (SRS). These methods rely heavily on offline simulations todetermine good control parameters, such as peak power. This isoperationally burdensome and error prone.

U.S. Pat. No. 9,438,369 describes increasing capacity by optimizationafter nonlinear modeling. U.S. Pat. No. 8,364,036 describes controllingoptical power within domains, and exchanging state information betweendomains. U.S. Pat. No. 8,781,317 describes methods to measure phasenonlinearities. U.S. Pat. No. 7,894,721 describes global optical controlwhere receiver changes are correlated to network perturbations. U.S.Pat. No. 7,457,538 describes performance monitoring using theanalog-to-digital converter of the receiver. U.S. Pat. No. 7,376,358describes location-specific monitoring of nonlinearities. U.S. Pat. No.7,356,256 describes digital monitoring along the optical line. US PatentPublication No. 2016/0315711 describes controlling the optical spectraldensity in a section.

SUMMARY

Through the latest innovations, optical networks are capable ofdynamically changing optical paths, and flexible transceivers arecapable of changing modulation formats and other transmissionparameters. In this environment, optical line control that provides goodperformance, scalability, and self-optimization is desirable.

Adjustment of one or more control parameters of a section of an opticalfiber network involves taking measurements of optical signals in thesection, deriving estimated data from the measurements and fromknowledge of the section, where the estimated data is a function ofoptical nonlinearity and of amplified spontaneous emission, and applyingone or more control algorithms using the estimated data to adjust theone or more control parameters.

BRIEF DESCRIPTION OF THE DRAWINGS AND APPENDIX

FIG. 1 illustrates a method for adjustment of control parameters insection of an optical fiber network;

FIG. 2 illustrates an example section of an optical fiber network;

FIG. 3 illustrates an example concave value function of excess margin;

FIG. 4 illustrates a first derivative of the example concave valuefunction;

FIG. 5 illustrates an example optical fiber network;

FIG. 6 illustrates a method for adjustment of control parametersaffecting the relative per-channel launch power and either the totallaunch power or the total received optical power; and

Appendix A is an example calculation of a cross-phase modulation (XPM)transfer function.

DETAILED DESCRIPTION

Optical network topologies can range from simple unamplifiedpoint-to-point, to branching chains of reconfigurable optical add dropmultiplexer (ROADM) sections, up to a full multi-connected mesh thatspans a continent.

In wavelength division multiplexing (WDM) systems, an optical fibernetwork connects wavelength selective switch (WSS) nodes via spans ofoptical fibers and optical amplifier devices. Pairs of flexible coherenttransceivers are connected over paths through the optical fiber network.Different channels are propagated through different paths in thenetwork. A flexible coherent transceiver can be reconfigured allowingtransmission parameters, for example, modulation scheme, to be adaptedto the selected path.

Some elements of the optical fiber network have the ability to do somelevel of per-channel power control. Such elements may include, forexample, the transmitter portions of the flexible coherent transceivers,a variable optical attenuator (VOA) under software control, and opticalequalizers. In another example, per-channel power is controllable byprovisioning a wavelength selective switch (WSS) node with loss values.A WSS node can have switching capabilities and per-channel powercontrol.

Some elements of the optical fiber network have the ability to do somelevel of total power control. Such elements may include, for example,optical amplifier devices. For example, the gain of an optical amplifierdevice is controllable by provisioning the optical amplifier device witha target gain. Equivalently, the total output power (TOP) of an opticalamplifier device is controllable by provisioning the optical amplifierdevice with a target total output power.

Some optical amplifier devices also have the ability to do some level ofper-channel power control, by provisioning the optical amplifier devicewith a target gain tilt profile. For simplicity, this document focuseson the following control parameters of a section of an optical fibernetwork: the loss values of a WSS node, which affect the launch powersof the signals launched into the optical fibers, and the target gainvalues (or target TOP values) of optical amplifier devices.

FIG. 1 is a flowchart illustration of a method for adjustment of controlparameters in a section of an optical fiber network. A section maycomprise most or all of the optical fiber network. If the optical fibernetwork is small, the section may indeed comprise all of the network.However, it is generally advantageous for the method to control a singlepoint-to-point section of optical amplifier devices and spans of opticalfiber between two nodes that contain ROADM, WSS, or other switchinghardware that may be present.

At 2, measurements of optical signals are taken at various locations inthe section. The measurements may include per-channel optical power(also referred to as power spectral density, especially in a flexiblegrid system) and total output power. For example, an optical powermonitor (OPM) device is able to measure per-channel optical power byswitching the optical connection to its input. Due to the cost of an OPMdevice, there is generally not an OPM device at each optical amplifierdevice. Taps and photodiodes may be placed, for example, at the inputand at the output of the optical amplifier devices. Each photodiode isoperative to measure the total optical power at the location of the tap.At locations where there is an OPM device and a tap and photodiode, themeasurement of total optical power may be used to calibrate theper-channel optical power measured by the OPM device.

At 4, estimated data is derived from the measurements, from the targetvalues, and from knowledge of the section and its components. Theestimated data may include, for example, the estimated per-channeloptical power at the output of the optical amplifier devices, theestimated incremental amplified spontaneous emission (ASE) powerintroduced by the optical amplifier devices, and the estimatedself-phase modulation (SPM) and cross-phase modulation (XPM) varianceaccumulated in the section. The estimated data may be derived using amodeling engine that models the propagation of signals through thecomponents of the section. Alternatively, the estimation ofnonlinearities and noise may be derived from specific measurements ofparameters as described in U.S. Pat. No. 8,594,499, U.S. Pat. No.7,356,256, U.S. Pat. No. 6,128,111, U.S. Pat. No. 6,687,464, U.S. Pat.No. 6,839,523, U.S. Pat. No. 7,376,358, U.S. Pat. No. 6,072,614, U.S.Pat. No. 6,064,501, and U.S. Pat. No. 5,513,029.

The estimated data is then used in a control algorithm to adjust thecontrol parameters. Various control algorithms are contemplated. Forexample, the control algorithm may make use of gradients and slew-ratelimited steepest descent. At 6, gradients of an objective function areevaluated, using the measurements and the estimated data. The values ofthe gradients are inaccurate, for at least the reason that themeasurements are noisy, the knowledge of the section and its componentsmay be inaccurate or incomplete, the modeling engine is inaccurate, andthe estimated data is inaccurate. Some of the channels propagatedthrough the section carry live traffic. That is, some of the channelsare in-service channels carrying traffic for customers. It is thereforeimportant not to adjust the components of the section in a manner thatwould jeopardize or degrade or destabilize the in-service channels.

At 8, the values of the gradients are used in steepest descentalgorithms to adjust control parameters of the section by a small stepin a direction of optimization of the objective function. That is, smalladjustments are applied to target values such as loss values of a WSSnode and the target gain (or target total output power) of an opticalamplifier device. Steepest descent algorithms are known to be noisetolerant and to be very safe if small steps are taken. The values ofsome control parameters that are adjusted may be set points foralgorithms that control other control parameters. For example, a valueof a per-channel optical power out of a WSS node may be a set point foran algorithm that adjusts the loss of the relevant pixels of that WSSnode. A total power may be a set point for an algorithm that adjuststotal gain, which may be a set point for a digital control loop whichadjusts a VOA loss and pump currents. A pump current may be a set pointfor an analog loop which adjusts a Field Effect Transistor (FET) bias.

The method illustrated in FIG. 1 may be repeated over the lifetime ofuse of the optical fiber network. For example, the method may berepeated every few seconds for 25 years. It is not necessary that allcontrol parameters be adjusted in each iteration of the method. Variouschanges occur over time, yielding updated measurement data, updatedestimated data, updated values for the gradients, an updated directionof optimization of the objective function, and updated adjustments tothe control parameters.

The optical fiber network may be partitioned into sections arbitrarily.For simplicity, this document focuses on an example section that enablestransmission of a set of optical signals along a particular transmissiondirection from a first WSS node to a second WSS node. (Signals are alsodirected along the opposite transmission direction, where the roles ofingress and egress are reversed. However, so as not to obscure thedescription of the technology, transmission along that oppositedirection is not illustrated and is not discussed in this document.)

FIG. 2 illustrates an example section 10 of an optical fiber network. Aningress WSS node 12 is connected to an egress WSS node 14 via spans 16of optical fiber. The length of a span 16 of optical fiber is typicallyin the range of approximately 80 km to approximately 100 km. The spans16 of optical fiber are coupled via optical amplifier devices 18. Anoptical pre-amplifier device 20 in the ingress WSS node 12 is opticallycoupled to the first span 16 of optical fiber. An optical pre-amplifierdevice 20 in the egress WSS node 14 is optically coupled to the finalspan 16 of optical fiber. One can index the spans 16 and the optical(pre-)amplifier devices 18,20 by an index j, with N representing thetotal number of spans of optical fiber coupling the ingress WSS node 12to the egress WSS node 14. For example, one can refer to the opticalpre-amplifier device 20 in the ingress WSS node 12 as the first opticalamplifier device, whose output is launched into the first span ofoptical fiber. Similarly, the output of the optical amplifier device jis launched into the span j of optical fiber.

As discussed above, the measurements taken at various locations in thesection may include per-channel optical power (also referred to as powerspectral density, especially in a flexible grid system) measured by OPMdevices and total output power measured by photodiodes. In the examplesection 10, OPM devices 22 at the ingress WSS node 12 and at the egressWSS node 14 are able to measure per-channel optical power across thespectrum at the output of the respective optical pre-amplifier device20. In the example section 10, taps and photodiodes are present at theinput and at the output of each optical (pre-)amplifier device 18,20 andare illustrated in FIG. 2 by small black squares. Each optical amplifierdevice 18 is comprised, together with its respective taps andphotodiodes and together with a shelf processor 24, in a network element26. For simplicity, only one such network element 26 is illustrated inFIG. 2. There is a shelf processor 28 comprised in the ingress WSS node12 and a shelf processor 30 comprised in the egress WSS node 14.

Each photodiode is operative to measure the total optical power at thelocation of its respective tap. At the output of the optical amplifierdevice j, the photodiode measures the total output power, which includesboth optical signal power and ASE power. The per-channel power measuredby the OPM device 22 is reliable only in terms of relative power acrossthe spectrum, because the loss along a cable 32 coupling the output ofthe optical pre-amplifier device 20 to the OPM device 22 is generallynot known. The total output power measured by the photodiode at theoutput of the optical pre-amplifier device 20 in the ingress WSS node 12can be used to calibrate the per-channel optical power measured by theOPM device 22, thus yielding a calibrated set of per-channel opticalpower measurements {P₁[i]}, where P₁[i] is the power of the channel ilaunched into the first span of optical fiber. The total output powermeasured by the photodiode at the output of the optical pre-amplifierdevice 20 in the egress WSS node 14 can be used to calibrate theper-channel optical power measured by the OPM device 22, thus yielding acalibrated set of per-channel optical power measurements {P_(N+1)[i]},where P_(N+1)[i] is the power of the channel i output from the opticalpre-amplifier device 20 in the egress WSS node 14. The integration ofper-channel optical power measurements to yield an aggregate power,comparison of the aggregate power to the measured total optical power,and calibration may be performed by firmware (not shown) in the WSS node12,14. Alternatively, the integration, comparison and calibration may beperformed by any other suitable firmware executed by a processor withinthe example section 10. Conventional optical power units are dBm. Inthis document, the per-channel optical power measurements {P_(j)[i]} areconveniently measured in units of Nepers relative to a Watt, because itis more convenient for the calculus of equations appearing hereinbelow.

A control system embedded in the section is operative to provisioncertain components of the section with specific target values. Forexample, the control system is operative to provision the ingress WSSnode 12 with loss values, and to provision the optical amplifier devices18 and the optical pre-amplifier devices 20 with respective target gainvalues or target TOP values. The control system comprises, for example,hardware (not shown) located in the ingress WSS node 12, hardware (notshown) located in the optical amplifier devices 18 and in the opticalpre-amplifier devices 20, and control firmware 34 executed by any one ofthe shelf processors within the section 10, for example, the shelfprocessor 30 comprised in the egress WSS node 14. The control firmware34 is stored in non-transitory computer-readable media coupled to theshelf processor.

In an alternative implementation, the control firmware 34 is executed byan external processor (not shown) that is in communication with thecontrollable elements of the section. The external processor may belocated in a physical server or may be virtualized as part of a cloudinfrastructure. The apparatus in which the external processor is locatedmay also store the control firmware 34 in non-transitorycomputer-readable media that is accessible by the external processor.

As discussed above, estimated data is derived from the measurements,from the target values, and from knowledge of the section and itscomponents. The estimated data may be derived using a modeling enginethat models the propagation of signals through the components of thesection.

The knowledge of the section and its components may include “knowncharacteristics”. Manufacturers and/or distributors of the componentsmay provide some of the known characteristics. Other knowncharacteristics may be determined by testing and/or calibrating thecomponents. Still other known characteristics may be provided byinspection of the section. The known characteristics may include, forexample, the topology of the section, one or more optical amplifiercharacteristics such as amplifier type (e.g. Erbium-doped fiberamplifier (EDFA), distributed Raman amplifier, lumped Raman amplifier),noise figure, ripple, spectral hole burning, and Total Output Power(TOP) limit, and one or more optical fiber characteristics such as fibertype, span length, nonlinear coefficients, effective area, losscoefficients, total loss, chromatic dispersion, and Stimulated RamanScattering (SRS).

Measured data (raw and/or calibrated), control data, and (optionally)known characteristics, are communicated within the section 10 over anoptical service channel (OSC), also known as an optical supervisorychannel. The WSS nodes 12,14 and the network elements 26 each comprisecircuitry 36 to support the OSC.

The modeling engine models the propagation of signals through componentsof the section. Specifically, the modeling engine employs fiber modelsfor the spans 16 of optical fiber in the section 10 and employsamplifier models for the optical (pre-)amplifier devices 18,20. Modelingfirmware 38 that uses the modeling engine is executed by any one of theshelf processors within the section 10, for example, the shelf processor24 comprised in the network element 26. The estimated data derived bythe modeling engine may include, for example, the estimated per-channeloptical power {P_(j)[i]} at the output of the optical amplifier j, whereP_(j)[i] is the power, measured in units of Nepers, of the channel ilaunched into the span j of optical fiber, and the estimated incrementalASE power {ASE_(j)[i]} at the output of the optical amplifier j. Themodeling engine may employ known techniques to derive the powerevolution of the optical signals through the section and to derive theincremental ASE power.

The accuracy of the estimated per-channel optical power at each of thefiber interfaces is important. Stimulated Raman Scattering (SRS) mayimpart in the range of approximately 1 dB to approximately 2 dB powertilt across the C band (1525 nm to 1565 nm) and in the range ofapproximately 3 dB to approximately 4 dB power tilt across the L band(1565 nm to 1610 nm). These power tilts may accumulate between spanswhere there is no WSS node to equalize the tilts. Channels at differentoptical powers experience very different optical degradation in terms ofASE (at low channel power) and optical nonlinearities (at high channelpower). Good modeling of the SRS tilt per span of optical fiber is partof what contributes to accurate estimated per-channel optical powers andaccurate estimated incremental ASE powers.

Once the estimated per-channel optical powers are of sufficient accuracy(which could be determined, for example, by comparing the estimatedper-channel optical powers for the output of the optical pre-amplifierdevice 20 in the egress WSS node 14 with the calibrated set ofper-channel optical power measurements {P_(N+1)[i]}), the modelingengine may derive the estimated self-phase modulation (SPM) andcross-phase modulation (XPM) variance accumulated in the section 10. Theestimated data is thus a function of optical nonlinearity and of ASE.

The modeling engine may model nonlinear interactions within the spans 16of optical fiber as Gaussian noise, as described in P. Poggiolini, “TheGN Model of Non-Linear Propagation in Uncompensated Coherent OpticalSystems”, Journal of Lightwave Technology, Vol. 30, No. 24, Dec. 15,2012; P. Poggiolini et al. “The GN Model of Fiber Non-Linear Propagationand its Applications”, Journal of Lightwave Technology, Vol. 32, No. 4,Feb. 14, 2014. Alternatively, the modeling engine may employ a differentmodel of the nonlinear interactions, for example, full non-linearSchrodinger Equation solutions using Fast Fourier transform (FFT) orfinite difference methods.

As described above, gradients of an objective function are evaluated,using the measurements and the estimated data.

In one aspect, the goal of the objective function is to minimize thetotal degradation through the section. Optimization of this objectivefunction minimizes a weighted sum of ratios of the total noise powerfrom ASE and optical nonlinearities to the power of the optical signals.This objective function is suitable for systems where there is nosoftware connection to convey information from the receiver modem to thesection.

An example objective function V₁ for a section, with the goal ofminimizing the total degradation through the section, is given inEquations (1) and (2):

$\begin{matrix}{V_{1} = {{\sum\limits_{i = 1}^{N_{CH}}{{C\lbrack i\rbrack}{{LNSR}\lbrack i\rbrack}}} = {\sum\limits_{i = 1}^{N_{CH}}{\sum\limits_{j = 1}^{N}{{C\lbrack i\rbrack}{{LNSR}_{j}\lbrack i\rbrack}}}}}} & (1) \\{{{LNSR}_{j}\lbrack i\rbrack} = {\frac{A\; S\; {E_{j}\lbrack i\rbrack}}{e^{P_{j}{\lbrack i\rbrack}}} + {\sum\limits_{k = 1}^{N_{CH}}{{{NL}_{j}\lbrack {i,k} \rbrack}e^{2{P_{j}{\lbrack k\rbrack}}}}}}} & (2)\end{matrix}$

In Equation (1), LNSR[i] denotes the incremental line noise-to-signalratio (NSR) for the channel i in the section, which can be expressed asa weighted sum over spans j of optical fiber of the incremental line NSRfor the channel i in the span j, denoted LNSR_(j)[i]. C[i] is aweighting value for the channel i to optionally bias the objectivefunction for particular higher-value signals. N_(CH) denotes the numberof channels in the signals in the section, and N denotes the number ofspans j of optical fiber in the section. C[i] may be a customer-definedweighting value. Alternatively, C[i] may be defined in a differentmanner. For example, when C[i]=Baudrate[i] the objective function V₁will converge to control parameters that maximize the capacity-bandwidthproduct in the optical fiber network. In another example, whenC[i]=Baudrate[i]×SNR[i] where SNR[i] is an estimate of thesignal-to-noise ratio (SNR) in linear units at the receiver modem whosechannel i traverses the section, the objective function V₁ will convergeto control parameters that maximize the capacity of the optical fibernetwork.

In Equation (2), P_(j)[i] is the power of the channel i at the output ofthe optical amplifier j, which is launched into the optical fiber of thespan j, ASE_(j)[i] is the incremental ASE power on the channel i at theoutput of the optical amplifier j, and NL_(j)[i, k] is the SPM/XPMnonlinear coefficient for Kerr interactions between the channel i andthe channel k at the span j. The power P_(j)[i] is measured in units ofNepers relative to a Watt. The second term in Equation (2) is asummation over all channel powers that impact the LNSR of the channel iat the span j. Where nonlinear interactions in one span are independentof nonlinear interactions in another span, the local optimum of thisobjective function V₁ is the global optimum. The paper I. Roberts, J. M.Kahn, D. Boertjes, “Convex Channel Power Optimization in Nonlinear WDMSystems using Gaussian Noise Model”, Journal of Lightwave Technology,Vol. 34, No. 13, Jul. 1, 2016 proves that the second term in Equation(2) is a convex function in the power P_(j)[i] when assuming a Gaussiannoise nonlinearity model.

The following discussion derives the gradients of the example objectivefunction V₁, which are evaluated to provide a direction for adjustmentof control parameters. A gradient vector ∇V_(1j) for control of theingress WSS node 12 is derived. A gain gradient for TOP control of theoptical amplifier devices 18 is derived.

The example objective function V₁ given in Equation (1) can be expressedas the sum over all spans j in the section of an example span objectivefunction V_(1j), which is given in Equation (3):

V _(1j)=Σ_(i=1) ^(N) ^(CH) C[i]LNSR _(j) [i]  (3)

The partial derivative of the example span objective function V_(1j)with respect to channel power for channel i in the span j of opticalfiber is given by Equation (4):

$\begin{matrix}{\frac{\partial V_{1j}}{\partial{P_{j}\lbrack i\rbrack}} = {{\sum\limits_{k = 1}^{N_{CH}}{{C\lbrack k\rbrack}\frac{\partial{{LNSR}_{j}\lbrack k\rbrack}}{\partial{P_{j}\lbrack i\rbrack}}}} = {{{- {C\lbrack i\rbrack}}\frac{A\; S\; {E_{j}\lbrack i\rbrack}}{e^{P_{j}{\lbrack i\rbrack}}}} + {\sum\limits_{k = 1}^{N_{CH}}{2{C\lbrack k\rbrack}{{NL}_{j}\lbrack {i,k} \rbrack}e^{2{P_{j}{\lbrack i\rbrack}}}}}}}} & (4)\end{matrix}$

where the channel power P_(j)[i] is fixed and the sum is over XPM/SPMterms over all channels dependent on channel power P_(j)[i].

A gradient vector ∇V_(1j) for a span j comprises the partial derivative

$\frac{\partial V_{1j}}{\partial{P_{j}\lbrack i\rbrack}}$

for each channel i from 1 to N_(CH). The partial derivative of theexample objective function V₁ with respect to WSS loss for the channel kin the section is given by Equation (5):

$\begin{matrix}{\frac{\partial V_{1}}{\partial{P\lbrack k\rbrack}} = {{\sum\limits_{j = 1}^{N}\frac{\partial V_{1j}}{\partial{P_{j}\lbrack k\rbrack}}} = {\sum\limits_{j = 1}^{N}\lbrack {{{- {C\lbrack k\rbrack}}\frac{A\; S\; {E_{j}\lbrack k\rbrack}}{e^{P_{j}{\lbrack k\rbrack}}}} + {\sum\limits_{i = 1}^{N_{CH}}{2{C\lbrack k\rbrack}{{NL}_{j}\lbrack {i,k} \rbrack}e^{2{P_{j}{\lbrack i\rbrack}}}}}} }}} & (5)\end{matrix}$

where P[k] is the power of the channel k out of the ingress WSS node 12which affects all spans in the section.

A gradient vector ∇V₁ for the section comprises the partial derivative∂V₁/∂P[k] for each channel k from 1 to N_(CH). The gradient vector ∇V₁can be evaluated from the customer values C[i], the knowncharacteristics NL_(j)[i, k], and the measured or estimated data{P_(j)[i]} and {ASE_(j)[i]}.

The incremental ASE power on the channel i induced by the opticalamplifier device j is given by the well known Equation (6):

ASE _(j) [i]=h*v*B _(e)(NF _(j) [i]*G _(j) [i]−1)  (6)

where h is Planck's constant, v is the optical frequency, B_(e) is theelectrical bandwidth of the noise filtering in the receiver, NF_(j)[i]is the noise figure for the channel i of the optical amplifier device j,and G_(j)[i] is the gain for the channel i of the optical amplifierdevice j.

The gradient of the ratio of the ASE power to the signal power term withrespect to gain G_(j)[i] is given by Equation (7):

$\begin{matrix}{{\nabla_{G_{j}{\lbrack i\rbrack}}( \frac{A\; S\; {E_{j}\lbrack i\rbrack}}{e^{P_{j}{\lbrack i\rbrack}}} )}\; \approx \; {- \frac{h \cdot v \cdot {{NF}_{j\; + \; 1}\lbrack i\rbrack} \cdot B_{e}}{e^{P_{j\; + \; 1}^{IN}{\lbrack i\rbrack}}}}} & (7)\end{matrix}$

where P_(j+1) ^(IN)[i] is the channel power at the input to the nextoptical amplifier device. This gradient is approximately the negative ofthe ratio of the incremental ASE power to signal power of the nextoptical amplifier device.

The gain gradient for the optical amplifier device j, averaged over allwavelengths, can be formed as given by Equation (8):

$\begin{matrix}{\frac{\partial V_{1\; j}}{\partial G_{j}} = {\sum\limits_{k\; = \; 1}^{N_{CH}}\lbrack {{{- {C\lbrack k\rbrack}}\; \frac{A\; S\; {E_{j\; + \; 1}\lbrack k\rbrack}}{e^{P_{j\; + \; 1}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {2\; {C\lbrack i\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}} & (8)\end{matrix}$

The gain gradient for the optical amplifier device j, averaged over allwavelengths, can be evaluated from the customer values C[i], the knowncharacteristics NL_(j)[i, k], and the measured or estimated data{P_(j)[i]} and {ASE_(j)[i]}.

As described above, the values of the gradients are used in steepestdescent algorithms to adjust control parameters of the section by asmall step in a direction of optimization of the objective function.

Small adjustments are applied to loss values of a WSS node and to targetTOP values of optical amplifier devices. The steepest descent algorithmis applied to the WSS node while assuming that the gains of the opticalamplifier devices are fixed. The steepest descent algorithm is appliedto all of the optical amplifier devices in parallel while assuming thatthe WSS pixel drive values are fixed.

For example, two loops may be run in parallel with a decoupling factor,as expressed in the vector Equation (9), Equation (10) and Equation(11):

$\begin{matrix}{{WSS\_ PowerTarget}_{NEW} = {{WSS\_ PowerTarget} - ( {\frac{MAXSTEP}{\max ( {\nabla V_{1}} )}*\lbrack {{\nabla V_{1}} - {{mean}( {\nabla V_{1}} )}} \rbrack} )}} & (9) \\{{TOP\_ Target}_{NEW} = {{TOP\_ Target} - {{{sign}\lbrack \frac{\partial V_{1\; j}}{\partial G_{j}} \rbrack}*0.1*{MAXSTEP}}}} & (10) \\{\mspace{79mu} {{{{if}\mspace{14mu} {TOP\_ Target}_{NEW}} \geq {TOP}_{LIMIT}},\mspace{79mu} {{{set}\mspace{14mu} {TOP\_ Target}_{NEW}} = {TOP}_{LIMIT}}}} & (11)\end{matrix}$

where WSS_PowerTarget_(NEW) and WSS_PowerTarget have values for eachchannel k from 1 to N_(CH), the decoupling factor in this example is0.1, and the target TOP for the optical amplifier device j is subject toan upper limit. An example MAXSTEP is 0.2 dB.

TOP control is used to decouple incremental SNR optimization in thissection from changes occurring in other sections of the optical fibernetwork. The subtraction of the change in average power (which isdenoted mean(∇V₁) in Equation (9) but is not quite equal to the averageof the changes) keeps the WSS output power constant, and the powerlaunched into the first span is controlled by the TOP of the firstamplifier. This scaling is important when one or more of the amplifiersreaches the limit of their TOP and can provide no more power. With thisscaling, the allocation of that limited power between the wavelengths iscleanly optimized.

Note also that there is no reliance on a second derivative for stepsize, and that this simple algorithm is robust to noise.

As mentioned above, where nonlinear interactions in one span areindependent of nonlinear interactions in another span, the local optimumof this objective function V₁ is the global optimum. Operationally, thispermits the adjustment of the control parameters for one section to beperformed in parallel to the adjustment of the control parameters forother sections of the optical fiber network. For example, the two loopsexpressed in the vector Equation (9), Equation (10) and Equation (11)may be run in parallel independently for several sections of the opticalfiber network.

In another aspect, the goal of the objective function is to maximize thecapacity or the reliability or both of a network by allocating margin tochannels that are at higher risk of failure at their designatedcapacities by taking away margin from channels with plenty of margin.This objective function is suitable for systems where, for at least somechannels, there is a software connection to convey information to thesection (or to the external processor) from the receiver modem thatreceives that channel. This objective function can also be used toprotect channels in service while trialing a new channel to see if itcan sustain a particular high capacity. Another value of this objectivefunction is to assist channels that are experiencing a slowlow-probability degradation event such as polarization dependent loss(PDL) by improving this weakened channel's line SNR at the expense ofother channels that have higher margin.

An arbitrary concave value function ƒ is introduced that takes as itsargument the excess margin SNR_(M)[i] on the channel i as determined atthe receiver modem. A positive value for SNR_(M)[i] indicates that totalSNR (including ASE, nonlinear effects, and internal receiver modemnoise) currently experienced by the channel i exceeds the SNR requiredfor error-free communications on that channel. A negative value forSNR_(M)[i] indicates that the total SNR currently experienced by thechannel i is less than the SNR required for error-free communications onthat channel. The concave value function ƒ (SNR_(M)[i]) expresses theutility of extra margin on a channel and whether the channel is betteroff sharing its excess margin. FIG. 3 illustrates an example concavevalue function ƒ having desirable properties, and FIG. 4 illustrates afirst derivative ƒ¹ of the example concave value function. It is scaledso that ƒ(0)=0 and ƒ¹(0)=1. The example concave value function ƒ isneutral (has a value of zero) for zero excess margin, decreases rapidlyfor negative excess margin, and increases then quickly plateaus forpositive excess margin.

An example objective function V₂ for a section that incorporatesinformation from the receiver modem is the sum over all controllablechannels of a concave value function ƒ of the excess margin, as given inEquations (12) and (13):

$\begin{matrix}{V_{2}\; = \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {{C\lbrack i\rbrack}\; {D\lbrack i\rbrack}\; {f( {{SNR}_{M}\lbrack i\rbrack} )}}}} & (12) \\{{S\; N\; {R_{M}\lbrack i\rbrack}}\; = \; {{- 10}\; {\log ( \frac{{LNSR}_{M}\lbrack i\rbrack}{{BLNSR}_{M}\lbrack i\rbrack} )}}} & (13)\end{matrix}$

In Equation (12), C[i] is a customer-defined weighting value for thechannel i to optionally bias the objective function for particularhigher-value signals. D[i] is a metric that is a function of thegeographic distance travelled by the channel i from the transmitter tothe receiver, or other such network value. Adjusting the function for Dallows an adaptation of the trade-off between the use of an opticalfiber (if optical fiber on this route is a scarce resource andinstalling or acquiring rights to more would be very expensive, then,for example, set D[i]=1), and minimizing the cost of the transceivers(if optical fiber is plentiful, then, for example, setD[i]=distance[i]).

In Equation (13), LNSR_(M)[i] is the line NSR for the channel i asmeasured at the receiver modem, and BLNSR_(M)[i] is a budgeted line NSRwhich factors in margin, implementation noise, and target Required Noiseto Signal Ratio (RNSR), required for the modem to be error free undernominal conditions, from the capacity commitment and forward errorchannel (FEC) performance for the channel i.

There are many different ways in which the budgeted line NSR for thechannel i, BLNSR_(M)[i], can be defined. For example, the budgeted lineNSR may be defined as given in Equation (14):

$\begin{matrix}{{{BLNSR}_{M}\lbrack i\rbrack}\; = \; {\frac{1}{m_{P}}\; ( {{{FEC\_ NSR}\lbrack i\rbrack}\; - \; {{INSR}\lbrack i\rbrack}\; - \; m_{A}} )}} & (14)\end{matrix}$

In Equation (14), FEC_NSR[i] is the NSR for the modulation format forthe channel i at the FEC threshold, INSR[i] is the modem implementationnoise for the channel i, m_(P) is a multiplicative margin applied to theline NSR, and m_(A) is an additive noise margin.

The example objective function V₂ given in Equation (12) can beexpressed as the sum over all channels i of an example channel objectivefunction V₂[i], which is given in Equation (15):

V ₂ [i]=C[i]D[i]ƒ(SNR _(M) [i])  (15)

The partial derivative of the example channel objective function V₂[i]with respect to channel power for channel i in the span j of opticalfiber is given by Equations (16), (17) and (18):

$\begin{matrix}{\frac{\partial{V_{2}\lbrack i\rbrack}}{\partial{P_{j}\lbrack i\rbrack}}\; = \; {{{A\lbrack i\rbrack}\; \frac{\partial}{\partial{P_{j}\lbrack i\rbrack}}\; ( {\sum\limits_{j\; = \; 1}^{N}\; {{LNSR}_{j}\lbrack i\rbrack}} )}\; = \; {{A\lbrack i\rbrack}\; \frac{\partial{{LNSR}_{j}\lbrack i\rbrack}}{\partial{P_{j}\lbrack i\rbrack}}}}} & (16) \\{{A\lbrack i\rbrack}\; = \; \frac{{- 10}\; {C\lbrack i\rbrack}\; {D\lbrack i\rbrack}\; {f^{1}( {S\; N\; {R_{M}\lbrack i\rbrack}} )}}{2.30\; {{LNSR}_{M}\lbrack i\rbrack}}} & (17) \\{{{LNSR}_{M}\lbrack i\rbrack}\; = \; {\sum\limits_{j\; = \; 1}^{N}\; {{LNSR}_{j}\lbrack i\rbrack}}} & (18)\end{matrix}$

Equations (16) and (17) demonstrate the use of the chain rule in thepartial derivative, and introduce the concept of a modem coefficientA[i] that encapsulates receiver modem information. The modem coefficientA[i] multiplies the partial derivative of the noise (LNSR) to powerratio of a specific section.

Through proper scaling of the metric D[i], the modem coefficient A[i]can be made equal to the first derivative of the example concave valuefunction ƒ: A[i]=ƒ¹(SNR_(M)[i]). The receiver modem whose channel itraverses the section is capable of determining the value of the modemcoefficient A[i]. For channels where receiver modem information isunavailable, the modem coefficient A[i] can be set to equal the number1.

The value of this proper scaling of the metric D[i] is to ground theexample concave value function ƒ of measured margin onto the exampleobjective function V₁ given in Equation (1) which can be shown to eithermaximize capacity or the capacity-product depending on the choice of theweighting value C[i]. When the modem coefficient A[i] is set to equalthe number 1 for all channels, the derivative in Equation (16) for theexample objective function V₂ is identical to the derivative in Equation(4) for the example objective function V₁. Thus the example concavevalue function ƒ which tends to help channels with less margin at theexpense of channels with more margin will operate around the controlparameters that are close to either maximizing capacity or thecapacity-distance product of the optical network.

The following discussion derives the gradients of the example objectivefunction V₂, which are evaluated to provide a direction for adjustmentof control parameters. A gradient vector ∇V_(2j) for control of theingress WSS node 12 is derived. A gain gradient for TOP control of theoptical amplifier devices 18 is derived.

By comparing Equation (16) and Equation (3), it is apparent that thegradients derived for the example objective function V₁ are applicableto the example objective function V₂, with the insertion of the modemcoefficient A[i]. In cases where the modem coefficient A[i] equals 1 forall channels, the gradients derived for the example objective functionV₁ are identical to the gradients derived for the example objectivefunction V₂.

The partial derivative of the example function V₂ with respect to WSSloss for the channel k in the section is therefore given by Equation(19):

$\begin{matrix}{\frac{\partial V_{2}}{\partial{P\lbrack k\rbrack}} = {\sum\limits_{j\; = \; 1}^{N}\lbrack {{{- {A\lbrack i\rbrack}}\; {C\lbrack k\rbrack}\; \frac{A\; S\; {E_{j}\lbrack k\rbrack}}{e^{P_{j}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{k\; = \; 1}^{N_{CH}}\; {2\; {A\lbrack i\rbrack}\; {C\lbrack k\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}} & (19)\end{matrix}$

where A[i]=ƒ¹(SNR_(M)[i]).

A gradient vector ∇V₂ for the section comprises the partial derivative∂V₂/∂P[k] for each channel k from 1 to N_(CH). The gradient vector ∇V₂can be evaluated from the modem coefficients A[i], the customer valuesC[i], the known characteristics NL_(j)[i, k], and the measured orestimated data {P_(j)[i]} and {ASE_(j)[i]}.

The gain gradient for the span j of optical fiber, averaged over allwavelengths, can be formed as given by Equation (20):

$\begin{matrix}{\frac{\partial V_{2\; j}}{\partial G_{j}} = {\sum\limits_{k\; = \; 1}^{N_{CH}}\lbrack {{{- {A\lbrack i\rbrack}}\; {C\lbrack k\rbrack}\; \frac{A\; S\; {E_{j\; + \; 1}\lbrack k\rbrack}}{e^{P_{j\; + \; 1}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {2\; {A\lbrack i\rbrack}\; {C\lbrack i\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}} & (20)\end{matrix}$

where A[i]=ƒ¹(SNR_(M)[i]).

The gain gradient for the optical amplifier device j, averaged over allwavelengths, can be evaluated from the modem coefficients A[i], thecustomer values C[i], the known characteristics NL_(j)[i, k], and themeasured or estimated data {P_(j)[i]} and {ASE_(j)[i]}.

The values of the gradients are used in steepest descent algorithms toadjust control parameters of the section by a small step in a directionof optimization of the objective function.

Small adjustments are applied to loss values of a WSS node and to targetTOP values of optical amplifier devices. The steepest descent algorithmis applied to the WSS node while assuming that the gains of the opticalamplifier devices are fixed. The steepest descent algorithm is appliedto all of the optical amplifier devices in parallel while assuming thatthe WSS pixel drive values are fixed.

For example, two loops may be run in parallel with a decoupling factor,as expressed in the vector Equation (21), Equation (22) and Equation(23):

$\begin{matrix}{{WSS\_ PowerTarget}_{NEW} = {{WSS\_ PowerTarget} - ( {\frac{MAXSTEP}{\max ( {\nabla V_{2}} )}*\lbrack {{\nabla V_{2}} - {{mean}( {\nabla V_{2}} )}} \rbrack} )}} & (21) \\{{TOP\_ Target}_{NEW} = {{TOP\_ Target} - {{{sign}\lbrack \frac{\partial V_{2\; j}}{\partial G_{j}} \rbrack}*0.1*{MAXSTEP}}}} & (22) \\{\mspace{79mu} {{{{if}\mspace{14mu} {TOP\_ Target}_{NEW}} \geq {TOP}_{LIMIT}},\mspace{79mu} {{{set}\mspace{14mu} {TOP\_ Target}_{NEW}} = {TOP}_{LIMIT}}}} & (23)\end{matrix}$

where WSS_PowerTarget_(NEW) and WSS_PowerTarget have values for eachchannel k from 1 to N_(CH), the decoupling factor in this example is0.1, and the target TOP for the optical amplifier device j is subject toan upper limit. An example MAXSTEP is 0.2 dB.

TOP control is used to decouple incremental SNR optimization in thissection from changes occurring other sections of the optical fibernetwork. The subtraction of the change in average power (which isdenoted mean(∇V₂) in Equation (21) but is not quite equal to the averageof the changes) keeps the WSS output power constant, and the powerlaunched into the first span is controlled by the TOP of the firstamplifier. This scaling is important when one or more of the amplifiersreaches the limit of their TOP and can provide no more power. With thisscaling, the allocation of that limited power between the wavelengths iscleanly optimized.

Note also that there is no reliance on a second derivative for stepsize, and that this simple algorithm is robust to noise.

When the ability for rapid introduction of new channels is desired,idlers may be used to pre-allocate the effects of those channels.

An ASE idler is treated by the first aspect (example objective functionV₁) as any other channel, with the appropriate XPM generatorcoefficient. The channel weight could be set to a static value of one.In the second aspect (example objective function V₂), once a modemsignal is switched to replace this ASE, then the margin from that modemwould be used to calculate the new weight in the usual way. Thediminished default value is used again after the ASE is switched backin.

To not cause the XPM from ASE idlers, a limited number of virtual idlerscan be used. Virtual idlers are treated just like ASE idlers, exceptthat their XPM generator coefficient is set to equal that of themodulation expected to be used. Virtual idlers do not consume photons,so the TOP limits need to be reduced by the virtual wattage.

A Boolean acceptance criterion should be used to decide on the choice ofa virtual idler versus an ASE idler in order to limit the SRS impact oftheir sudden conversion to real signals, assuming that all virtualidlers are allowed to switch at once. Define A to be the integral ofvirtual power spectral density out of the WSS, across a 1 THz intervalcentered at the wavelength of the candidate virtual idler, including thevirtual power of that candidate idler. Define B to be the integral ofreal power spectral density across the same 1 THz interval centered atthe wavelength of the candidate virtual idler. Choose epsilon to be asmall positive number to avoid division by zero, e.g. 100 microWatts.The virtual idler is acceptable if A/(B+epsilon)<0.25.

In yet another aspect, the objective function is a combination of theabove two objective functions. For example, the example objectivefunction is given by Equation (24):

V ₃ =V ₂ −V ₁=Σ_(i=1) ^(N) ^(CH) C[i]D[i]ƒ(SNR _(M) [i])−Σ_(i=1) ^(N)^(CH) C[i]LNSR[i]  (24)

With this example objective function V₃, the goal of the objectivefunction is to balance the goals of minimizing the total degradationthrough the section with maximizing capacity or reliability or both ofthe optical fiber network by re-allocating margin among the channelsthat are propagated through the section. The discussion above ofderiving gradients and applying the gradients in steepest descentalgorithms is applicable also to the example objective function V₃.

Returning now to FIG. 2, consider how this example section 10 could bemodified to independently amplify different bands of transmission. Forexample, the section 10 could simultaneously handle the C band (1525 nmto 1565 nm) and the L band (1565 nm to 1610 nm). The ingress WSS node 12could have two independent WSS filters to control individual channelpowers for the C band and the L band, respectively. Each of the optical(pre-)amplifier devices 18,20 could be replaced by a set of two optical(pre-)amplifier devices, one for the C band and one for the L band. TheC-band channels and the L-band channels propagate through the same spans16 of optical fiber, where there is fiber nonlinear interaction betweenall the channels. That is, the nonlinear interaction in the spans ofoptical fiber is across all channels being propagated, including C-bandchannels and L-band channels. There is also strong SRS which makes forsignificant power differences between channels compared to the case of asingle band, given that the SRS is approximately proportional to thesquare of the optical bandwidth.

In a variation of the first aspect, the example objective function V₁applies to the full set of channels in the C band and the L band.

The partial derivative of the example objective function V₁ with respectto C-band WSS loss for the channel k in the section is given by Equation(25), where the channel k is in the C band:

$\begin{matrix}{\frac{\partial V_{1}}{\partial{P\lbrack k\rbrack}} = {{\sum\limits_{j\; = \; 1}^{N}\frac{\partial V_{1\; j}}{\partial{P_{j}\lbrack k\rbrack}}} = {\sum\limits_{j\; = \; 1}^{N}\lbrack {{{- {C\lbrack k\rbrack}}\; \frac{A\; S\; {E_{j}\lbrack k\rbrack}}{e^{P_{j}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {2\; {C\lbrack k\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}}} & (25)\end{matrix}$

where P[k] is the power of the channel k out of the ingress WSS node 12which affects all spans in the section.

The partial derivative of the example objective function V₁ with respectto L-band WSS loss for the channel k in the section is given by Equation(26), where the channel k is in the L band:

$\begin{matrix}{\frac{\partial V_{1}}{\partial{P\lbrack k\rbrack}} = {{\sum\limits_{j\; = \; 1}^{N}\frac{\partial V_{1\; j}}{\partial{P_{j}\lbrack k\rbrack}}} = {\sum\limits_{j\; = \; 1}^{N}\lbrack {{{- {C\lbrack k\rbrack}}\; \frac{A\; S\; {E_{j}\lbrack k\rbrack}}{e^{P_{j}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {2\; {C\lbrack k\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}}} & (26)\end{matrix}$

where P[k] is the power of the channel k out of the ingress WSS node 12which affects all spans in the section.

In Equation (25), the summation of the nonlinear interaction is over allN_(CH) channels i in the C band and in the L band. In Equation (26), thesummation of the nonlinear interaction is over all N_(CH) channels i inthe C band and in the L band.

A gradient vector ∇V₁(C) for the section for the C band comprises thepartial derivative ∂V₁/∂P[k] for each channel k in the C band from 1 toN_(CH) ^(C). A gradient vector ∇V₁(L) for the section for the L bandcomprises the partial derivative ∂V₁/∂P[k] for each channel k in the Lband from 1 to N_(CH) ^(L). The gradient vectors ∇V₁(C) and ∇V₁(L) canbe evaluated from the customer values C[i], the known characteristicsNL_(j)[i, k], and the measured or estimated data {P_(j)[i]} and{ASE_(j)[i]}.

The gain gradient for the optical amplifier device j, averaged over allwavelengths in the C band, can be formed as given by Equation (27):

$\begin{matrix}{\frac{\partial{V_{1\; j}(C)}}{\partial G_{j}}\; = \; {\sum\limits_{k\; = \; 1}^{N_{CH}^{C}}\lbrack {{{- {C\lbrack k\rbrack}}\; \frac{A\; S\; {E_{j\; + \; 1}\lbrack k\rbrack}}{e^{P_{j\; + \; 1}{\lbrack k\rbrack}}}}\; + \; {\sum\limits_{i\; = \; 1}^{N_{CH}}\; {2\; {C\lbrack i\rbrack}\; {{NL}_{j}\lbrack {i,\; k} \rbrack}\; e^{2\; {P_{j}{\lbrack i\rbrack}}}}}} \rbrack}} & (27)\end{matrix}$

where the outer summation is over the channels k in the C band, and theinner summation is over all N_(CH) channels i in the C band and the Lband.

The gain gradient for the optical amplifier device j, averaged over allwavelengths in the L band, can be formed as given by Equation (28):

$\begin{matrix}{\frac{\partial{V_{1j}(L)}}{\partial G_{j}} = {\sum\limits_{k = 1}^{N_{CH}^{L}}\lbrack {{{- {C\lbrack k\rbrack}}\frac{A\; S\; {E_{j + 1}\lbrack k\rbrack}}{e^{P_{j + 1}{\lbrack k\rbrack}}}} + {\sum\limits_{i = 1}^{N_{CH}}{2{C\lbrack i\rbrack}{{NL}_{j}\lbrack {i,k} \rbrack}e^{2{P_{j}{\lbrack i\rbrack}}}}}} \rbrack}} & (28)\end{matrix}$

where the outer summation is over the channels k in the L band, and theinner summation is over all N_(CH) channels i in the C band and the Lband. The total number of channels in the C band and the L band, denotedN_(CH), is the sum of the number of channels in the C band, denotedNL_(H), and the number of channels in the L band, denoted N_(CH) ^(L).That is, N_(CH)=N_(CH) ^(C)+N_(CH) ^(L).

The gain gradients can be evaluated from the customer values C[i], theknown characteristics NL_(j)[i, k], and the measured or estimated data{P_(j)[i]} and {ASE_(j)[i]}.

Small adjustments are applied to loss values of a WSS node and to targetTOP values of optical amplifier devices. The steepest descent algorithmis applied to the WSS node while assuming that the gains of the opticalamplifier devices are fixed. The steepest descent algorithm is appliedto all of the optical amplifier devices in parallel while assuming thatthe WSS pixel drive values are fixed.

For example, four loops may be run in parallel with a decoupling factor,as expressed in the vector Equations (29) and (30), Equations (31) and(32) and Equations (33) and (34):

$\begin{matrix}{{{WSS\_ PowerTarget}_{NEW}(C)} = {{{WSS\_ PowerTarget}(C)} - ( {\frac{MAXSTEP}{\max ( {\nabla{V_{1}(C)}} )}*\lbrack {{\nabla{V_{1}(C)}} - {{mean}( {\nabla{V_{1}(C)}} )}} \rbrack} )}} & (29) \\{{{WSS\_ PowerTarget}_{NEW}(L)} = {{{WSS\_ PowerTarget}(L)} - ( {\frac{MAXSTEP}{\max ( {\nabla{V_{1}(L)}} )}*\lbrack {{\nabla{V_{1}(L)}} - {{mean}( {\nabla{V_{1}(L)}} )}} \rbrack} )}} & (30) \\{{{TOP\_ Target}_{NEW}(C)} = {{{TOP\_ Target}(C)} - {{{sign}\lbrack \frac{\partial{V_{1j}(C)}}{\partial G_{j}} \rbrack}*0.1*{MAXSTEP}}}} & (31) \\{\mspace{79mu} {{{{if}\mspace{14mu} {TOP\_ Target}_{NEW}(C)} \geq {{TOP}_{LIMIT}(C)}},\mspace{79mu} {{{set}\mspace{14mu} {TOP\_ Target}_{NEW}(C)} = {{TOP}_{LIMIT}(C)}}}} & (32) \\{{{TOP\_ Target}_{NEW}(L)} = {{{TOP\_ Target}(L)} - {{{sign}\lbrack \frac{\partial{V_{1j}(L)}}{\partial G_{j}} \rbrack}*0.1*{MAXSTEP}}}} & (33) \\{\mspace{79mu} {{{{if}\mspace{14mu} {TOP\_ Target}_{NEW}(L)} \geq {{TOP}_{LIMIT}(L)}},\mspace{79mu} {{{set}\mspace{14mu} {TOP\_ Target}_{NEW}(L)} = {{TOP}_{LIMIT}(L)}}}} & (34)\end{matrix}$

where WSS_PowerTarget_(NEW)(C) and WSS_PowerTarget(C) have values foreach channel k in the C band from 1 to N_(CH) ^(C),WSS_PowerTarget_(NEW)(L) and WSS_PowerTarget(L) have values for eachchannel k in the L band from 1 to N_(CH) ^(L), the decoupling factor inthis example is 0.1, and the target TOP for the optical amplifier deviceis subject to an upper limit (dependent on the band). An example MAXSTEPis 0.2 dB.

In a variation of the second aspect, the example objective function V₂applies to the full set of channels in the C band and the L band.Similar equations and loops can be derived for that case, for example,by replacing the customer values C[i] in Equations (25) through (34)with the product of the customer values C[i] and the modem coefficientsA[i].

For clarity, the examples apply a Gaussian nonlinearity noise model. Themethods described in the document can be used where other models ofoptical nonlinear interactions provide a better representation, such aswhere the nonlinearities are not substantially independent betweenspans.

In some of the methods described above a model is used to approximatethe nonlinear characteristics of the optical system, such as theGaussian Noise Model. Models make certain simplifying assumptions thatmay not apply to a given system. The Gaussian Noise Model assumes thatthe nonlinear optical components are Gaussian and are independentbetween optical spans. This is a reasonable approximation when there issignificant chromatic dispersion in the fibers and there is no opticaldispersion compensation. On systems with low chromatic dispersion orwith optical dispersion compensation, such as older undersea cables,this approximation is not valid. The nonlinear effects can bepredominantly angular rotations as opposed to being additive and equalin all directions. The nonlinear effect can be substantially correlatedbetween spans, which causes their total variance to grow more stronglyalong the line than the power-addition that corresponds to the case ofsummation of uncorrelated values.

Commercial situations exist where the cable and line amplifiers weremanufactured and installed by one organization, and the modems aresupplied to the owner of the cable by a competing organization. Here,the parameters of the optical line may not be accurately made availableto the supplier of the modems nor to the owner of the cable. Opticalpowers along the line may not be measured, or may not be accuratelycommunicated. Losses may not be known. Fiber types and characteristicsmay not be communicated. Without accurate parameters, models may not beaccurate.

Optical signals owned by one organization may be routed over a sectionof optical line that is the responsibility of another organization,inhibiting communication of accurate optical parameters. Softwareboundaries, incompatibilities, control zones, administrative regions,and such can also inhibit communication.

In the methods described above it is often desirable to controldistinctly within each ROADM section, and to control each amplifierpower. That segmentation may not be feasible or desirable, and soend-to-end control might be desired.

The rest of this document describes an alternate method for adjustmentof control parameters in an optical fiber network. This alternate methodis suitable for use in networks where there is low chromatic dispersionon the optical line, for example, where the average chromatic dispersionis less than 5 ps/nm/km, and is suitable for use in networks whereoptical dispersion compensation, such as dispersion-compensated opticalfibers, is employed. This alternate method is suitable for use innetworks where the fiber types are not necessarily known. This alternatemethod is suitable for use in networks where the per-channel powersoutput from the optical amplifier devices in the optical line are notnecessarily known.

FIG. 5 illustrates an example optical fiber network 50. A first flexiblecoherent transceiver 52 is connected to a second flexible coherenttransceiver 54 via an optical line 55. The precise nature of the opticalline 55 is not necessarily known. In a simplest implementation (notshown), the optical fiber network 50 has a simple unamplifiedpoint-to-point topology, and the optical line 55 consists of a singlespan of optical fiber. In other implementations, the optical line 55comprises multiple spans 56 of optical fiber that are coupled viaoptical amplifier devices 58. In this case, the optical fiber network 50can be described as a multi-span optical fiber network. In someexamples, the multiple spans 56 of optical fiber and the opticalamplifier devices 58 form a single path for all channels from thetransmitters 62 comprised in the first flexible coherent transceiver 52to receivers 64 comprised in the second flexible coherent transceiver54. For simplicity, this is the example used in much of thisdescription. In other examples, the optical line 55 comprisesconcatenated reconfigurable optical add drop multiplexer (ROADM)sections (not shown), and some of the channels transmitted by thetransmitters 62 may be branched off to transmitter portions of otherflexible coherent transceivers (not shown), and some of the channelsreceived by the receivers 64 may have been transmitted by transmittersof other flexible coherent transceivers (not shown). Typically, onetransmitter 62 connects to one receiver 64, but other topologies such asoptical multicast or drop-and-continue can be used. A pair of transmitand receive circuits are often located together on one board or module,and referred to as a modem 60. Other configurations include the transmitcircuits and the receive circuits each being separate, or a plurality oftransmit and/or receive circuits being physically located together.

The first flexible coherent transceiver 52 is operative to transmit anoptical signal composed of up to N_(CH) different channels, indexed byi, through the optical line 55. A wavelength selective switch (WSS)component 66 comprised in the first flexible coherent transceiver 52 isoperative to multiplex the outputs of N_(CH) transmitters 62. An opticalpre-amplifier device 70 is operative to amplify the multiplexed outputsto produce the optical signal. An optical power monitor (OPM) 71 deviceis able to measure per-channel optical power across the spectrum at theoutput of the optical pre-amplifier device 70. Each transmitter 62 isoperative to produce a modulated optical carrier for a respective one ofthe channels. The first flexible coherent transceiver 52 may compriseadditional components that, for the sake of simplicity, are notillustrated or discussed in this document.

The relative per-channel optical powers launched into the optical line55, also referred to as the launch power spectral density, especially ina flexible grid system, and also referred to as the relative per-channellaunch powers and denoted {P[i]}, are controllable by provisioning theWSS component 66 with loss values. The total optical power (TOP) of theoptical signal, also referred to as the total launch power, iscontrollable by provisioning the optical pre-amplifier device 70 with atarget gain or, equivalently, with a target total output power.

The second flexible coherent transceiver 54 is operative to receive anoptical signal composed of up to N_(CH) different channels, indexed byi, through the optical line 55. An optical pre-amplifier device 72 isoperative to amplify the received optical signal. A WSS component 74comprised in the second flexible coherent transceiver 54 is operative todemultiplex the amplified received optical signal into multiple signalsand to provide the multiple signals to N_(CH) receivers 64. Receiversgenerally detect the optical signal, decode the stream of symbols orbits, perform error correction, and provide a bit stream to anelectrical or optical interface such as a client signal or a backplane.The second flexible coherent transceiver 54 may comprise additionalcomponents that, for the sake of simplicity, are not illustrated ordiscussed in this document.

The total optical power of the optical signal that is received by thesecond flexible coherent transceiver 54 (referred to as “the totalreceived optical power”) is controllable by provisioning all of theoptical amplifier devices 58 in the optical line 55 with a common targettotal output power or, equivalently, with a fixed target gain. In manyconfigurations it is not possible or feasible to adjust the output powerof each optical amplifier device 58 individually, for example inundersea links. In other configurations, various subsets of the opticalamplifier devices 58 may be controlled together. For simplicity ofdescription here, we will use the basic example of common control forall optical amplifier devices 58 comprised in the optical line 55.

In this document, optical power measurements are conveniently measuredin units of Nepers relative to a Watt, because it is more convenient forthe calculus of equations appearing herein.

Cross Polarization Modulation, Stimulated Raman Scattering, BrillionScattering, and Four Wave Mixing are examples of other sources ofnonlinear interaction between optical signals, but, for simplicity inthis description, polarization is ignored and Cross Phase Modulation(XPM) is used in the descriptions.

Each receiver 64 is operative to measure the following quantities forits respective channel i: a line noise-to-signal ratio (NSR) denotedLNSR_(M)[i]; the accumulated cross-phase modulation (XPM) variance onchannel i due to all other channels relative to the signal power of thechannel i, denoted XPM[i]; the accumulated self-phase modulation (SPM)variance on channel i relative to the signal power of the channel i,denoted SPM[i]; and the accumulated amplified spontaneous emission (ASE)variance on channel i relative to the signal power of the channel i,denoted ASE[i].

Each receiver 64 is operative to determine the following quantities forits respective channel i: an excess margin, denoted SNR_(M)[i]; and abudgeted line NSR, denoted BLNSR_(M)[i]. A positive value for SNR_(M)[i]indicates that total SNR (including ASE, nonlinear effects, and internalreceiver modem noise) currently experienced by the channel i exceeds theSNR required for error-free communications on that channel. A negativevalue for SNR_(M)[i] indicates that the total SNR currently experiencedby the channel i is less than the SNR required for error-freecommunications on that channel. There are many different ways in whichthe budgeted line NSR for the channel i can be defined. One exampledefinition is provided above in Equation (14).

FIG. 6 is a flowchart illustration of a method for adjustment of controlparameters affecting the relative per-channel launch powers and thetotal power launched into an optical line.

At 82, the relative per-channel launch powers, denoted {P[i]}, aredetermined. In one example, the relative per-channel launch powers aremeasured, for example, by the OPM 71, at the first flexible coherenttransceiver 52. In another example, the relative per-channel launchpowers are determined from knowledge of the target loss values for theWSS component 66.

At 84, the receivers 64 make measurements and determine certainquantities. For example, for the channel i, the measurements include theline NSR, denoted LNSR_(M)[i]; the accumulated ASE variance relative tosignal power on the channel i, denoted ASE[i]; the accumulated SPMvariance relative to signal power on the channel i, denoted SPM[i]; andthe accumulated XPM variance relative to signal power on the channel idue to all other channels, denoted XPM[i]. For example, for the channeli, the determined quantities include the budgeted line NSR, denotedBLNSR_(M)[i]; and the excess margin, denoted SNR_(M)[i].

The measurements and quantities determined by the receivers 64 and therelative per-channel launch powers are used in a control algorithm thatadjusts control parameters. The control algorithm may adjust the lossvalues of the WSS component 66 of the first flexible coherenttransceiver 52 to affect the per-channel launch powers. The controlalgorithm may adjust a target gain or, equivalently, a target totaloutput power of the optical pre-amplifier device 70 to affect a totallaunch power. The control algorithm may adjust a target gain or,equivalently, a target total output power of all the optical amplifierdevices 58 in the optical line 55 to affect the total received opticalpower at the second flexible coherent transceiver 54 and/or at othertransceivers.

Various control algorithms are contemplated. For example, the controlalgorithm may result in a bounded step change adjustment to controlparameters. More specifically, as an example, the control algorithm maymake use of gradients and slew-rate limited steepest descent.

At 86, gradients of an objective function are evaluated, using themeasurements and quantities determined by the receivers 64 and therelative per-channel launch powers. The values of the gradients areinaccurate, for at least the reason that the measurements are noisy, andthe gradients are based on estimations and approximations. Some of thechannels propagated through the optical line carry live traffic. Thatis, some of the channels are in-service channels carrying traffic forcustomers. It is therefore important not to adjust the launch powers ina manner that would jeopardize or degrade or destabilize the in-servicechannels.

At 88, the values of the gradients are used in steepest descentalgorithms to adjust control parameters by a small step in a directionof optimization of the objective function. That is, small adjustmentsare applied to target values such as the loss values of the WSScomponent 66 of the first flexible coherent transceiver 52, a targetgain or, equivalently, a target total output power of the opticalpre-amplifier device 70, and a target gain or, equivalently, a targettotal output power of all the optical amplifier devices 58 in theoptical line 55. Steepest descent algorithms are known to be noisetolerant and to be very safe if small steps are taken.

The method illustrated in FIG. 6 may be repeated over the lifetime ofuse of the optical fiber network. For example, the method may berepeated every few seconds for 25 years. It is not necessary that allcontrol parameters be adjusted in each iteration of the method. Variouschanges occur over time, yielding updated margin information, updatedmeasurement data, updated values for the gradients, and updateddirection of optimization of the objective function, and updatedadjustments to the control parameters.

Returning briefly to FIG. 5, the control algorithm may be implemented ascontrol firmware 92 that is executed by an external processor 94 that isin communication with the controllable elements of the optical fibernetwork 50. The external processor 94 may be located in a physicalserver or may be virtualized as part of a cloud infrastructure. Anapparatus 96 in which the external processor 94 is located may store thecontrol firmware in non-transitory computer-readable media 98 that isaccessible by the external processor 94.

An example objective function is discussed hereinbelow. The exampleobjective function involves an arbitrary concave value function ƒ thattakes as its argument the excess margin SNR_(M)[i] on the channel i asdetermined at the receiver. The concave value function ƒ(SNR_(M)[i])expresses the utility of extra margin on a channel and whether thechannel is better off sharing its excess margin. As mentioned above,FIG. 3 illustrates an example concave value function ƒ having desirableproperties, and FIG. 4 illustrates a first derivative ƒ¹ of the exampleconcave value function. It is scaled so that ƒ(0)=0 and ƒ¹(0)=1. Theexample concave value function ƒ is neutral (has a value of zero) forzero excess margin, decreases rapidly for negative excess margin, andincreases then quickly plateaus for positive excess margin.

An example objective function V₄ is the sum over all controllablechannels of a concave value function ƒ of the excess margin, as given inEquations (35) and (13):

$\begin{matrix}{V_{4} = {\sum\limits_{i = 1}^{N_{CH}}{{C\lbrack i\rbrack}{f( {{SNR}_{M}\lbrack i\rbrack} )}}}} & (35) \\{{{SNR}_{M}\lbrack i\rbrack} = {{- 10}\; {\log ( \frac{{LNSR}_{M}\lbrack i\rbrack}{{BLNSR}_{M}\lbrack i\rbrack} )}}} & (13)\end{matrix}$

In Equation (35), C[i] is a customer-defined weighting value for thechannel i to optionally bias the objective function for particularhigher-value signals.

The goal of the example objective function V₄ is to maximize thecapacity or the reliability or both of a network by allocating margin tochannels that are at higher risk of failure at their designatedcapacities by taking away margin from channels with plenty of margin.This objective function is suitable for systems where, for at least somechannels, there is a software connection to convey information to anexternal processor from the receivers that receive those channels. Thisobjective function can also be used protect channels in service whiletrialing a new channel to see if it can sustain a particular highcapacity. Another value of this objective function is to assist channelsthat are experiencing a slow low-probability degradation event such aspolarization dependent loss (PDL) by improving this weakened channel'sline SNR at the expense of other channels that have higher margin.

The example objective function V₄ given in Equation (35) can beexpressed as the sum over all channels i of an example channel objectivefunction V₄[i], which is given in Equation (36):

V ₄ [i]=C[i]ƒ(SNR _(M) [i])  (36)

The partial derivative of the example channel objective function V₄[i]with respect to the launch power for channel k, denoted P[k], is givenby Equations (37) and (38):

$\begin{matrix}{\frac{\partial{V_{4}\lbrack i\rbrack}}{\partial{P\lbrack k\rbrack}} = {{{A\lbrack i\rbrack}\frac{\partial}{\partial{P\lbrack k\rbrack}}( {{LNSR}\lbrack i\rbrack} )} = {{A\lbrack i\rbrack}\frac{\partial{{LNSR}\lbrack i\rbrack}}{\partial{P\lbrack k\rbrack}}}}} & (37) \\{{A\lbrack i\rbrack} = \frac{{- 10}\; {C\lbrack i\rbrack}{f^{1}( {{SNR}_{M}\lbrack i\rbrack} )}}{2.30\; {{LNSR}_{M}\lbrack i\rbrack}}} & (38)\end{matrix}$

Equations (37) and (38) demonstrate the use of the chain rule in thepartial derivative, and introduce the concept of a modem coefficientA[i] that encapsulates receiver modem information. The modem coefficientA[i] multiplies the partial derivative of the line NSR on the channel i,LNSR[i], with respect to the launch power for channel k, denoted P[k].

The example objective function V₄ given in equation (35) is identical tothe example objective function V₂ given in equation (12) for the casewhere the metric D[i] is equal to one for all channels. The modemcoefficient A[i] given in equation (38) is identical to the modemcoefficient A[i] given in equation (17) for the case where the metricD[i] is equal to one for all channels. As mentioned above, D[i] is ametric that is a function of the geographic distance travelled by thechannel i from the transmitter to the receiver, or other such networkvalue. However, for simplicity in this case, D[i] is chosen to be equalfor the channels being considered.

The line NSR for the channel i, denoted LNSR[i], can be considered to bethe sum of the accumulated ASE variance relative to signal power, theaccumulated SPM variance relative to signal power, and the accumulatedXPM variance relative to signal power:

LNSR[i]=ASE[i]+SPM[i]+XPM[i]  (39)

The accumulated ASE variance relative to signal power on the channel k,denoted ASE[k], can be expressed as ASE[k]=N_(ASE)[k]/e^(P[k]), whereN_(ASE)[k] denotes the accumulated ASE on the channel k. The partialderivative of the accumulated ASE variance relative to signal power onthe channel k with respect to the launch power on the channel k can beexpressed as the negative of the accumulated ASE variance relative tosignal power on the channel k:

$\begin{matrix}{\frac{\partial{{ASE}\lbrack k\rbrack}}{\partial{P\lbrack k\rbrack}} = {{\frac{\partial}{\partial{P\lbrack k\rbrack}}( {{N_{ASE}\lbrack k\rbrack}/e^{P{\lbrack k\rbrack}}} )} = {{- \frac{N_{ASE}\lbrack k\rbrack}{e^{P{\lbrack k\rbrack}}}} = {- {{ASE}\lbrack k\rbrack}}}}} & (40)\end{matrix}$

The accumulated self-phase modulation (SPM) variance relative to signalpower on the channel k can be expressed as SPM[k]=κe^(2P[k]), where K isa constant value. The partial derivative of the accumulated SPM variancerelative to signal power on the channel k with respect to the launchpower on the channel k can be expressed as twice the accumulated SPMvariance relative to signal power on the channel k:

$\begin{matrix}{\frac{\partial{{SPM}\lbrack k\rbrack}}{\partial{P\lbrack k\rbrack}} = {{\frac{\partial}{\partial{P\lbrack k\rbrack}}( {\kappa \; e^{2\; {P{\lbrack k\rbrack}}}} )} = {{2\; \kappa \; e^{2\; {P{\lbrack k\rbrack}}}} = {2\; {{SPM}\lbrack k\rbrack}}}}} & (41)\end{matrix}$

The accumulated cross-phase modulation (XPM) variance relative to signalpower on the channel i due to all other channels can be expressed asfollows:

XPM[i]=κΣ _(k=1) ^(N) ^(CH) e ^(2P[k]) M[k]W[k,i], where i≠k  (42)

where κ is the constant value referred to above, and M[k] denotes thecoefficient of relative aggression of the modulation format used on thechannel k.

W[k, i] denotes the walk-off effect as a function of time offset due tothe chromatic dispersion of the full length of the optical line. Fornotational convenience, W[k, k]=0 to allow SPM to be handled separately.Let Δƒ[k, i] be the frequency difference between the carrier of thechannel k and the carrier of the channel i. The time offset due to thechromatic dispersion of the full length of the optical line may becalculated in units of the symbol interval on the channel k from thefrequency difference Δƒ[k, i]. The time offset may be denoted c[k, i].The walk-off effect can be expressed as a function of this time offset:W[k, i]=W(c[k, i]). Various methods may be used to determine values forthe XPM transfer function W[k, i]. An example method to determine valuesfor the XPM transfer function W[k, i] is described in Appendix A.

Given a measure of the accumulated XPM variance relative to signal poweron the channel i due to all other channels, denoted XPM[i], the sourceof that XPM variance from a given channel k can be reasonably allocatedas:

$\begin{matrix}\frac{{{XPM}\lbrack i\rbrack}e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}{\sum\limits_{k = 1}^{N_{CH}}{e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}} & (43)\end{matrix}$

The derivative of this quantity with respect to the launch power of thechannel k, denoted P[k], is given by:

$\begin{matrix}{\frac{\partial{{XPM}\lbrack i\rbrack}}{\partial{P\lbrack k\rbrack}} = \frac{2\; {{XPM}\lbrack i\rbrack}e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}{\sum\limits_{k = 1}^{N_{CH}}{e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}}} & (44)\end{matrix}$

with the approximation that the sum of powers is still constant in thedenominator.

Based on equations (40), (41) and (44), a matrix of the partialderivative of the line NSR for the channel i, denoted LNSR[i], withrespect to the launch power for channel k, denoted P[k], can beevaluated as follows:

$\begin{matrix}{\frac{\partial{{LNSR}\lbrack i\rbrack}}{\partial{P\lbrack k\rbrack}} = \{ \begin{matrix}{{{2\; {{SPM}\lbrack k\rbrack}} - {{ASE}\lbrack k\rbrack}},} & {{{where}\mspace{14mu} i} = k} \\{\frac{2\; {{XPM}\lbrack i\rbrack}e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}{\sum\limits_{k = 1}^{N_{CH}}{e^{2\; {P{\lbrack k\rbrack}}}{M\lbrack k\rbrack}{W\lbrack {k,i} \rbrack}}},} & {{{where}\mspace{14mu} i} \neq k}\end{matrix} } & (45)\end{matrix}$

Referring to equation (37), by evaluating the partial derivative of theline NSR for the channel i, denoted LNSR[i], with respect to the launchpower for channel k, the partial derivative of the example channelobjective function V₄[i] can be evaluated.

A gradient vector ∇V₄ comprises the partial derivative ∂V₄/∂P[k] foreach channel k from 1 to N_(CH). Referring to equations (37) and (45)the gradient vector ∇V₄ can be evaluated from the modem coefficientsA[i], the per-channel launch powers {P[i]}, the measured accumulated ASEvariance relative to signal power denoted {ASE[i]}, the measuredaccumulated SPM variance relative to signal power denoted {SPM[i]}, andthe measured accumulated XPM variance relative to signal power denoted{XPM[i]}.

The partial derivative of the example objective function V₄ with respectto the total launch power, denoted TOP, is given by Equations (46) and(38):

$\begin{matrix}{\frac{\partial V_{4}}{\partial{TOP}} = {{\sum\limits_{i = 1}^{N_{CH}}{{C\lbrack i\rbrack}\frac{\partial}{\partial{TOP}}( {f( {{SNR}_{M}\lbrack i\rbrack} )} )}} = {\sum\limits_{i = 1}^{N_{CH}}{{A\lbrack i\rbrack}\frac{\partial{{LNSR}\lbrack i\rbrack}}{\partial{TOP}}}}}} & (46) \\{{A\lbrack i\rbrack} = \frac{{- 10}\; {C\lbrack i\rbrack}{f^{1}( {{SNR}_{M}\lbrack i\rbrack} )}}{2.30\; {{LNSR}_{M}\lbrack i\rbrack}}} & (38)\end{matrix}$

The partial derivative of the line NSR for the channel i, denotedLNSR[i], with respect to the total launch power, denoted TOP, may bemeasured by launching a sequence of total power values and measuring theresulting line NSR after each launch. An example sequence is {TOP+ε,TOP+ε, TOP-ε, TOP-ε, TOP-ε, TOP-ε, TOP+ε, TOP+ε} in dB for an arbitrarypositive constant c, resulting in the following line NSR measurements{LNSR[i]₁, LNSR[i]₂, LNSR[i]₃, LNSR[i]₄, LNSR[i]₅, LNSR[i]₆, LNSR[i]₇,LNSR[i]₈}. The phase of this dipole set (that is, whether the positiveconstant E is added to or is subtracted from the total launch power TOP)should follows a pseudorandom pattern between sets so as to notcorrelate with any fast environmental factors. The dipole structure isdesigned to be very tolerant to slow environmental ramps. From thisdipole set, the partial derivative of the line NSR for the channel i,denoted LNSR[i], with respect to the total launch power, denoted TOP,can be calculated as follows:

$\begin{matrix}{\frac{\partial{{LNSR}\lbrack i\rbrack}}{\partial{TOP}} = {\frac{1}{8\; ɛ}( {{\sum_{+ ɛ}{{LNSR}\lbrack i\rbrack}_{j}} - {\sum_{- ɛ}{{LNSR}\lbrack i\rbrack}_{j}}} )}} & (47)\end{matrix}$

where the sum of line NSR measurements for the channel i resulting fromsubtracting the positive constant c from the total launch power isitself subtracted from the sum of line NSR measurements for the channeli resulting from adding the positive constant ε to the total launchpower. The optimum value of the positive constant ε that produces theminimum margin variance can be determined from Markov Chain analysis ofthe stepping algorithm, with a quadratic expression for margin versusTOP.

Using equations (38), (46) and (47), the partial derivative of theexample objective function V₄ with respect to the total launch power canbe evaluated.

The values of the gradients are used in steepest descent algorithms toadjust control parameters of the section by a small step in a directionof optimization of the objective function.

Small adjustments are applied to loss values of the WSS component 66 ofthe first flexible coherent transceiver 52. Small adjustments areapplied either to a target TOP value (or equivalently, to a target gain)for the optical pre-amplifier device 70 or to a common target TOP value(or equivalently, to a fixed target gain) for all of the opticalamplifier devices 58. In order to separate the effects of per-channellaunch powers from the total launch power (or the total received power),which can be limited, the per-channel launch powers are rescaled aftereach gradient step. The total launch power (or the total received power)can be separately controlled by a stepping algorithm.

For example, two loops may be run in parallel with a decoupling factor,as expressed in the vector Equation (48), Equation (49) and Equation(50):

$\begin{matrix}{{WSS\_ PowerTarget}_{New} = {{WSS\_ PowerTarget} - ( {\frac{MAXSTEP}{\max ( {\nabla V_{4}} )}*\lbrack {{\nabla V_{4}} - {{mean}( {\nabla V_{4}} )}} \rbrack} )}} & (48) \\{{TOP\_ Target}_{NEW} = {{TOP\_ Target} - {{{sign}( \frac{\partial V_{4}}{\partial{TOP}} )}*0.1*{MAXSTEP}}}} & (49) \\{{{{if}\mspace{14mu} {TOP\_ Target}_{NEW}} \geq {TOP}_{LIMIT}},{{{set}\mspace{14mu} {TOP\_ Target}_{NEW}} = {TOP}_{LIMIT}}} & (50)\end{matrix}$

where WSS_PowerTarget_(NEW) and WSS_PowerTarget have values for eachchannel k from 1 to N_(CH), the decoupling factor in this example is0.1, and the target TOP for the optical pre-amplifier device 70 (or forall the optical amplifier devices 58) is subject to an upper limit. Anexample MAXSTEP is 0.2 dB.

In vector equation (48), mean(∇V₄) is the logarithm of the mean of theper-channel power changes e^(∇V) ⁴ ^([i]), expressed as follows:

mean(∇V ₄)=ln(Σ_(i=1) ^(N) ^(CH) e ^(P[i]+∇V) ⁴ ^([i]))−ln(Σ_(i=1) ^(N)^(CH) e ^(P[i)])  (51)

The quantity mean(∇V₄) is not equal to the mean of the elements of∇V₄[i], especially when the power levels differ.

Note also that there is no reliance on a second derivative for stepsize, and that this simple algorithm is robust to noise.

Note that this alternate method does not assume that the opticalnonlinear products are Gaussian nor independent between spans. Forexample, this alternate method is suitable for use in multi-span opticalfiber networks where optical nonlinear interactions, for example,self-phase modulation (SPM) and/or cross-phase modulation (XPM), are notcompletely independent from one span to another. That is, opticalnonlinear interactions are at least partially dependent from one of thespans to another one of the spans. Knowledge of the fiber types, losses,and optical powers are not required. This alternate method can be usedin systems where simplified models are not accurate, or where thecontrol method does not have accurate knowledge of the parameters of theline.

For clarity, the examples use ASE from Erbium doped fiber amplifiers(EDFAs). Other power or gain dependent degradations such asdouble-Raleigh scattering from Raman amplifiers may be included in theestimations.

Modern high capacity optical transmission systems use coherent modems(also known as coherent transceivers). The techniques described in thisdocument may also be used with other kinds of optical transmitters andreceivers.

The description shows specific examples of objective functions and thederivation of control algorithms from those objective functions. Otherobjective functions may be used. Using a convex objective function andderiving a control algorithm from an objective function is convenient.However, other control algorithms and methods may be used.

Measurements by the receivers 64 of various optical parameters have beendescribed, namely the line NSR, denoted LNSR_(M)[i]; the accumulated ASEvariance relative to signal power on the channel i, denoted ASE[i]; theaccumulated SPM variance relative to signal power on the channel i,denoted SPM[i]; and the accumulated XPM variance relative to signalpower on the channel i due to all other channels, denoted XPM[i]. Theseparameters can be measured in other ways. The gradients can becalculated or estimated from other parameters, such as from the changein LNSR[k] at a receiver as a function of a dither or test patternapplied to each channel launch power P[i].

As an alternative to the gradient descent method described in thisdocument, direct calculation of the desired operating point can be doneby using algebra on the parameter values. The Newton method could beused for aggressive optimization. Combinations of calculation, modellingand measurement can be used.

For simplicity, the control algorithms described in this document havesole control of the control parameters of the section. Other controlalgorithms or provisioning or constraints may also be present or active.

The scope of the claims should not be limited by the details set forthin the examples, but should be given the broadest interpretationconsistent with the description as a whole.

APPENDIX A

Various methods may be used to determine values for the XPM transferfunction W[k, i]. For example, where a Gaussian-Noise (GN) approximationof the per-span XPM noise-to-signal ratio is appropriate, the XPMtransfer function W[k, i] can be approximated as follows:

${{W\lbrack {k,i} \rbrack} \approx {{\frac{1}{\ln (3)} \cdot \frac{B\lbrack i\rbrack}{B\lbrack k\rbrack}}{\ln ( \frac{{2{{k - i}}} + 1}{{2{{k - i}}} - 1} )}}},$

where |k−i|≥1This expression holds for dispersion compensated systems. Here

${B\lbrack i\rbrack} = \frac{\alpha_{i}\gamma_{i}^{2}L_{{eff},i}^{2}S_{i}T_{i}^{2}}{\beta_{i}}$

whereλ_(i) is the wavelength of the channel i in units [m],α_(i)=α(λ_(i)) is the fiber attenuation at the wavelength λ_(i) in units[1/m],

$\gamma_{i} = {\frac{8}{9} \cdot \frac{2\; \pi \; n_{2}}{A_{eff}\lambda_{i}}}$

is the fiber nonlinear parameter at the wavelength λ_(i) in units [1/m],

$L_{{eff},i} = {\frac{1}{\alpha_{i}}( {1 - e^{{- \alpha_{i}}L}} )}$

is the fiber effective length at the wavelength λ_(i) in units [m],

${\beta_{i}} = {10^{- 6}\frac{\lambda_{i}^{2}}{2\; \pi \; c}{D( \lambda_{i} )}}$

is the chromatic dispersion at the wavelength λ_(i) in units[s²/(rad m)],c is the speed of light in units [m/s],D (λ_(i)) is the dispersion at the wavelength λ_(i) in units [ps/nm/km],T_(i) is the symbol interval of the channel i in units [s],S_(i)=1/(T_(i)·Δw_(i)) is the spectral occupancy of the channeli[unitless], andΔw_(i) is the total bandwidth of the channel i including any guard band.

What is claimed is:
 1. A method of power control in a multi-span opticalfiber network, the method comprising: measuring optical characteristicsin multiple receivers comprised in a flexible coherent transceiver ofthe multi-span optical fiber network, each of the multiple receiversoperative to handle communications on a respective channel; andadjusting an optical power of a signal on each of the multiple channelsas a function of the optical characteristics, wherein for each of themultiple receivers, the optical characteristics include opticalnonlinear interactions on the respective channel, the optical nonlinearinteractions being at least partially dependent from one span to anotherspan.
 2. The method as recited in claim 1, wherein for each of themultiple receivers, the optical characteristics include accumulatedcross-phase modulation on the respective channel due to other channels.3. The method as recited in claim 1, wherein for each of the multiplereceivers, the optical characteristics include accumulated self-phasemodulation on the respective channel.
 4. The method as recited in claim1, wherein for each of the multiple receivers, the opticalcharacteristics include accumulated amplified spontaneous emission onthe respective channel.
 5. The method as recited in claim 1, wherein theoptical power that is adjusted is the optical launch power.
 6. Themethod as recited in claim 1, wherein adjusting the optical powerincludes adjusting loss values of a wavelength selective switch (WSS)component.
 7. The method as recited in claim 1, wherein adjusting theoptical power includes adjusting a total launch power.
 8. The method asrecited in claim 1, wherein adjusting the optical power includesadjusting a target total optical power or a target gain of an opticalpre-amplifier device comprised in another flexible coherent transceiverof the multi-span optical fiber network.
 9. The method as recited inclaim 1, wherein adjusting the optical power includes adjusting a totalreceived power at the flexible coherent transceiver.
 10. The method asrecited in claim 1, wherein adjusting the optical power includesadjusting a common target total optical power or a fixed target gain ofmultiple optical amplifier devices in the multi-span optical fibernetwork.
 11. The method as recited in claim 1, wherein adjusting theoptical power is a bounded step change.
 12. The method as recited inclaim 11, wherein adjusting the optical power includes: usingmeasurements of the optical characteristics to evaluate gradients of anobjective function; and determining a direction of the bounded stepchange from one or more dimensions of the gradients.
 13. The method asrecited in claim 12, wherein the objective function incorporates a valuefunction of margin determined by the multiple receivers.
 14. The methodas recited in claim 13, wherein the value function is a concave valuefunction.
 15. An apparatus comprising: a processor configured to receivemeasurements of optical characteristics from a flexible coherenttransceiver of a multi-span optical fiber network, the measurements ofoptical characteristics having been measured in multiple receiverscomprised in the flexible coherent transceiver, each of the multiplereceivers operative to handle communications on a respective channel,the processor configured to apply a control algorithm that adjusts anoptical power of a signal on each of the multiple channels as a functionof the optical characteristics, wherein for each of the multiplereceivers, the optical characteristics include optical nonlinearinteractions on the respective channel, the optical nonlinearinteractions being at least partially dependent from one span to anotherspan.
 16. The apparatus as recited in claim 15, wherein for each of themultiple receivers, the optical characteristics include accumulatedcross-phase modulation on the respective channel due to other channels.17. The apparatus as recited in claim 15, wherein for each of themultiple receivers, the optical characteristics include accumulatedself-phase modulation on the respective channel.
 18. The apparatus asrecited in claim 15, wherein the optical power that is adjusted is theoptical launch power.
 19. The apparatus as recited in claim 15, whereinthe control algorithm adjusts the optical power by adjusting a totallaunch power.
 20. The apparatus as recited in claim 15, wherein thecontrol algorithm adjusts the optical power by adjusting a totalreceived power at the flexible coherent transceiver.